Enough stalling, let’s get to the battle.
We must consider the two battles separately. In Part 1, we will analyze the first battle, of who will win the highest roll, and we will leave the analysis of the second battle, for the 2nd highest roll, for Part 2.
To calculate the probability of the attacker winning the highest roll, we will first count the permutations in which the defender achieves a highest roll of x and calculate how many of those permutations would result in an outright win for the defense, a tie, which goes to the defense, or a win for the attack.
For example, since, as calculated above, there is a 9/36 chance of the defense’s highest roll being 5, and there are a total of 6⁵ = 7776 permutations, clearly (9/36) * 7776 = 1944 of those permutations will yield a highest defense roll of 5. To win, the attack then needs to get a highest roll of 6, the probability of which is 91/216, as calculated above, so (91/216) * 1944 = 819 of the 1944 permutations which yielded a highest defense roll of 6 will result in a victory for attack. To achieve a tie, attack must roll a highest roll of 5, the probability of which is 61/216, so (61/216) * 1944 = 549 of those permutations will result in ties, and the remainder (1944–819–549 = 576) will result in outright defense wins.
We can make similar calculations for all possible outcomes for defense. See the below table.
We can then calculate the conditional² probability of a defense victory, of a tie, and of an attack victory, by dividing the number of permutations yielding the selected outcome (e.g. attack wins) by the count of the broader group of outcomes (e.g. defense rolls a highest roll of 5).
We can also visualize the conditional probabilities.
Chart 4 gives the same false impression that most people have initially, namely that attack has a big advantage overall. But this is because it ignores the probabilities of the highest defense rolls themselves. In fact, higher rolls are significantly more likely than lower rolls, as can be seen in Table 3.
For this reason, total probabilities are a more effective measure. It is even simpler to calculate the total probability of each outcome. We simply divide each permutation count by the total number of possible permutations, which is 7776.
We can then sum the total permutations which result in a victory for the attack (3667) and divide by the total possible permutations (7776) to obtain a win probability of 47.15% for the attack.
Below is a chart of total win probabilities by highest defensive roll. We include a tie as part of a defense victory for simplicity.
Finally, we will calculate the joint³ probabilities of each possible highest roll for both defense and attack. Since the highest roll for attack and defense are independent, we can simply multiply the probabilities together to obtain the joint probability. Note that in the below two graphics, red indicate victory for the attack, while blue denotes a defensive victory and pale blue indicates a tie, which goes to the defense.
We can also graph this data visually. Please note the configurations of the axes in the below chart, which have been configured to allow for maximal visibility.
